arXiv research

Reviewed and extended Wang-Yau quasi-local mass definitions.

The article calculates the near horizon limit of Wang--Yau quasi-local mass.

In this paper lower bounds are obtained for quasi-local masses in terms of charge, angular momentum, and horizon area. In particular we treat three quasi-local masses based on a Hamiltonian approach, namely the Brown-York, Liu-Yau, and Wang-Yau masses. The geometric inequalities are motivated by analogous results for t…

Paper discusses quasilocal mass and fill-ins, proving positivity and exploring definitions.

In this note, we compute the limit of the Wang-Yau quasi-local mass on unit spheres at spatial infinity of an asymptotically flat initial data set. Similar to the small sphere limit of the Wang-Yau quasi-local mass, we prove that the leading order term of the quasi-local mass recovers the stress-energy tensor. For a va…

In this article, we survey recent developments in defining the quasi-local mass in general relativity. We discuss various approaches and the properties and applications of the different definitions. Among the expected properties, we focus on the rigidity property: for a surface in the Minkowski spacetime, one expects t…

Inspired by the work of Chen-Zhang \cite{Chen-Zhang}, we derive an evolution formula for the Wang-Yau quasi-local energy in reference to a static space, introduced by Chen-Wang-Wang-Yau \cite{CWWY}. If the reference static space represents a mass minimizing, static extension of the initial surface $Σ$, we observe that …

Defines a new quasi-local mass related to spacetime harmonic functions.

In this paper, we show that the Chen-Nester-Tung (CNT) quasi-local energy is closely related to the Wang-Yau (WY) quasi-local mass. As a particular example, we compute the second variation of the CNT quasi-local energy for axially symmetric Kerr-like spacetimes with axially symmetric embeddings at the obvious critical …

Proposes a new quasi-local mass for timelike 2-surfaces in spacetimes.

We calculate the limits of the quasi-local angular momentum and center-of-mass defined by Chen-Wang-Yau \cite{CWY} for a family of spacelike two-spheres approaching future null infinity in an asymptotically flat spacetime admitting a Bondi-Sachs expansion. Our result complements earlier work of Chen-Wang-Yau \cite{CWY2…

Study limits of quasi-local angular momentum at infinity of gravitating systems.

We give a brief review of the definition of the Wang-Yau quasilocal mass and discuss the evaluation of which on surfaces of unit size at null infinity of an axi-symmetric spacetime in Bondi-van der Burg-Metzner coordinates.

New definitions of conserved quantities at null infinity resolve ambiguities in general relativity.

Study examines Wang-Yau quasi-local energy in strong fields near apparent horizons.

In this paper, we prove the following theorem regarding the Wang-Yau quasi-local energy of a spacelike two-surface in a spacetime: Let $Σ$ be a boundary component of some compact, time-symmetric, spacelike hypersurface $Ω$ in a time-oriented spacetime $N$ satisfying the dominant energy condition. Suppose the induced me…

We study a functional on the boundary of a compact Riemannian 3-manifold of nonnegative scalar curvature. The functional arises as the second variation of the Wang-Yau quasi-local energy in general relativity. We prove that the functional is positive definite on large coordinate spheres, and more general on nearly roun…

There are two important statements regarding the Trautman-Bondi mass at null infinity: one is the positivity, and the other is the Bondi mass loss formula, which are both global in nature. In this note, we compute the limit of the Wang-Yau quasi-local mass on unit spheres at null infinity of an asymptotically flat spac…

We discuss our recent work [4] in which gravitational radiation was studied by evaluating the Wang-Yau quasi-local mass of surfaces of fixed size at the infinity of both axial and polar perturbations of the Schwarzschild spacetime, à la Chandrasekhar [1].

There are two important statements regarding the Trautman-Bondi mass [1,8,5] at null infinity: one is the positivity [7,6], and the other is the Bondi mass loss formula [1], which are both global in nature. The positivity of the quasi-local mass can potentially lead to a local description at null infinity. This is conf…

In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang-Yau quasi-local mass for a surface in spacetime introduced in [7] and [8] is defined by minimizing quasi-local energy associated with admissible isometric embeddings of the …

We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consis…

Study proves positivity of quasi-local masses in general relativity using spinors.

Given a spacelike 2-surface $Σ$ in a spacetime $N$ and a constant future timelike unit vector $T_0$ in $\R^{3,1}$, we derive upper and lower estimates of Wang-Yau quasilocal energy $E(Σ, X, T_0)$ for a given isometric embedding $X$ of $Σ$ into a flat 3-slice in $\R^{3,1}$. The quantity $E(Σ, X, T_0)$ itself depends …

Researchers prove CWY angular momentum is supertranslation invariant in double null gauge.

In this article, we study the small sphere limit of the Wang-Yau quasi-local energy defined in [18,19]. Given a point $p$ in a spacetime $N$, we consider a canonical family of surfaces approaching $p$ along its future null cone and evaluate the limit of the Wang-Yau quasi-local energy. The evaluation relies on solving …

We study the quasi-local energy (QLE) and the surface geometry for Kerr spacetime in the Boyer-Lindquist coordinates without taking the slow rotation approximation. We also consider in the region $r\leq2m$, which is inside the ergosphere. For a certain region, $r>r_{k}(a)$, the Gaussian curvature of the surface with co…

The paper defines cross-section continuity for angular momentum definitions and finds the CWY definition valid.

The X-ADM mass is shown to be equivalent to the ADM mass, proving the X-positive mass theorem in all dimensions.

The paper studies nonlinear mass concepts in 3-manifolds with nonnegative scalar curvature.

Equivalence proven for isocapacitary mass notions.

The paper defines a new mass quantity for 3-manifolds and proves a positive mass theorem.

Introduce new boundary mass for asymptotically flat half-manifolds

We prove directly without using a density theorem that (i) the ADM mass defined in the usual way on an asymptotically flat manifold is equal to the mass defined intrinsically using Ricci tensor; (ii) the Hamiltonian formulation of center of mass and the center of mass defined intrinsically using Ricci tensor are the sa…

Continuous metrics on R^3 with specific properties have non-negative harmonic mass.

The paper examines mass aspects at future null infinity and limits of quasilocal mass.

Unified definition of mass aspect function for weakly regular hyperbolic manifolds.

Study the mass of flat 3-manifolds with boundary using specific methods.

Local mass perspective on Bayesian inference

On asymptotically flat and asymptotically hyperbolic manifolds, by evaluating the total mass via the Ricci tensor, we show that the limits of certain Brown-York type and Hawking type quasi-local mass integrals equal the total mass of the manifold in all dimensions.

Simple proof for sphere mass calculation.

New ADM mass definition for weakly regular manifolds.

New theorem for spacetime mass in noncompact regions.

New optimal transport method handles mass creation and destruction.

Study on residual Monge-Ampère mass for symmetric plurisubharmonic functions.

Huisken's isoperimetric mass is always nonnegative.

Study on residual Monge-Ampère mass for symmetric plurisubharmonic functions.

Surveying mass in 2D hyperbolic geometry, overcoming challenges via minimisation.